Time dilation

Let’s place the light clock on a train travelling at velocity $v$ relative to the station. We call RF train (RFT) the frame at rest with the train and RF platform (RFP) the frame of the station. The very same tick will be timed differently by the two observers, and it is precisely from this comparison that the relativity of time arises.

Seen from the train (RFT): the beam rises and falls in a straight line, exactly as before, and the tick lasts $\Delta\tau = 2d/c$. This is the proper time of the clock, measured by whoever travels with it.

Seen from the platform (RFP): while the beam rises, the train — and with it the upper mirror — moves sideways. The beam therefore traces a longer, zig-zag path:

The zig-zag path as seen from the platform. In one tick the train advances by $v\,\Delta t$, and the beam travels the hypotenuse: longer than the simple up-and-down path seen from the train.

Since light travels at $c$ also in the RFP (second postulate), covering a longer path takes more time. Applying Pythagoras’ theorem to the right triangle for half the path, whose legs are the height $d$ and the sideways displacement $v\,\Delta t/2$, and whose hypotenuse is the light path $c\,\Delta t/2$:

$\left(\frac{c\,\Delta t}{2}\right)^2 = d^2 + \left(\frac{v\,\Delta t}{2}\right)^2$

Substituting $d = c\,\Delta\tau/2$ and rearranging terms isolates $\Delta t$:

$(c^2 - v^2)(\Delta t)^2 = c^2(\Delta\tau)^2$

$\ev{\Delta t = \frac{\Delta\tau}{\sqrt{1 - v^2/c^2}} = \gamma\,\Delta\tau}$

where $\gamma = 1/\sqrt{1 - v^2/c^2}$ is the Lorentz factor.

Key formula

$\Delta t = \gamma\,\Delta\tau$ $\gamma = \frac{1}{\sqrt{1-v^2/c^2}} \geq 1$

Since $\gamma \geq 1$, we always have $\Delta t \geq \Delta\tau$: the tick of the moving clock, timed from the ground, lasts longer. In other words, the moving clock ticks more slowly.

Law — Time dilation

A clock moving relative to an observer runs more slowly than a clock at rest. More precisely, if two events occur at the same point in an observer’s RF (for example the tick and the tock of the train’s clock), in the RF of another observer moving relative to the first at velocity $v$ they are separated by a time $\Delta t = \gamma\,\Delta\tau > \Delta\tau$.

Proper time

The shortest time between two events is always the one measured in the reference frame in which the two events are co-spatial (occur at the same point). This time is called the proper time $\Delta\tau$.

No one can exceed the speed of light

If $v > c$ were possible, $\sqrt{1 - v^2/c^2}$ would be imaginary: $\Delta t$ would be a complex number, whereas we expect a real time. This mathematical impossibility has a deep meaning: no massive object can reach the speed of light. Only massless particles (photons, gluons) travel at $c$, and they do so exactly at $c$, in every reference frame.

Links

Topics: Special relativity Concepts: Time dilation Skills: Changing reference frame Methods: Minkowski diagram Objects: Light clock

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